The Costa Concordia, which is operated by Costa Cruises, had sailed from Civitavecchia near Rome on Friday on a regular weekly Mediterranean cruise when it ran aground.
The president of Costa Cruises, Gianni Onorato, said the main task for the company was now to assist passengers and crew, and help repatriate them.
He said it was difficult to determine what had happened, but that the ship had experienced a blackout after hitting "a big rock".
Mr Onorato added: "We will be working in full transparency with Italian authorities" to understand the causes of the disaster.
He said normal lifeboat evacuation had become "almost impossible" because the ship had listed so quickly.
Rescued passengers were initially accommodated in hotels, schools and a church on Giglio.
"Usually there are 700 people on the island at this time of year, so receiving 4,000 and some in the middle of the night wasn't easy," she said. "Some people jumped in the sea so they had hypothermia."
Some "tens" of British passengers are believed to have been on board, said the UK Foreign Office, which has sent a team to the area.
The Divers are searching rooms in the Costa Concordia, which is lying on its side 200m (650ft) off Giglio island.
It sank on the first night of a Mediterranean cruise on Friday. Most of its 4,000 passengers and crew reached land by lifeboats but some swam ashore.
Three people died. Late on Saturday, two people were found alive on board.
Rescuers have reached a man and a woman stranded in a cabin two decks down on the half-submerged ship, according to Italian media reports.
The precise number of those who remain unaccounted for is unclear. Late on Saturday local official Giuseppe Linardi said up to 41 people were missing.
